Large time behaviour of solutions to the 3D Navier–Stokes equations with damping

Abstract

In this paper, we consider large time behaviour of solutions to the 3D Navier–Stokes equations with damping term $$|u|^{\beta -1}u$$ ( $$\beta \ge 1$$ ). For $$\beta >\frac{7}{3}$$ , we derive decay rates of the $$L^2$$ -norm of the solutions. By using Fourier splitting method, we also prove that the solutions are asymptotically equivalent to the solutions of the classic 3D Navier–Stokes equations.